US2024412456A1PendingUtilityA1
Convolutional neural networks on tetrahedral meshes
Est. expiryJun 7, 2043(~16.8 yrs left)· nominal 20-yr term from priority
G06T 17/20G06T 3/40
56
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Claims
Abstract
Disclosed herein are methods for graphing convolutional neural networks in tetrahedral meshes. In some embodiments, the methods include computing a volumetric Laplace Beltrami Operator. In some embodiments, the methods include feeding the LBO into a neural network. In some embodiments, the methods include down-sampling a tetrahedral mesh.
Claims
exact text as granted — not AI-modifiedWhat is claimed is:
1 . A method of using a tetrahedral mesh on a neural network, comprising:
computing a volumetric Laplace Beltrami operator (LBO) for the tetrahedral mesh; feeding into the neural network the LBO and a set of signals for each vertex of the neural network; and down-sampling the tetrahedral mesh.
2 . The method of claim 1 , wherein computing the volumetric LBO is performed based on the following function:
Δ
f
(
v
i
,
)
=
1
d
i
∑
j
∈
N
(
i
)
k
i
,
j
(
f
(
v
i
)
-
f
(
v
j
)
)
where N(i) includes the adjacent vertices of vertex v i , and d i is total tetrahedral volume of adjacent tetrahedra to vertex v i , and k i,j is the string constant.
3 . The method of claim 1 , wherein the set of signals comprises at least one input signal and at least one output signal.
4 . The method of claim 3 , wherein the at least one input signal is defined as:
x in ∈R N .
5 . The method of claim 4 , wherein the at least one output signal is defined as:
x out ∈R M
6 . The method of claim 5 , wherein a convolution is defined as:
𝔵
o
u
t
=
g
*
T
𝔵
i
n
=
Φ
(
(
Φ
T
g
)
⊙
(
Φ
τ
𝔵
in
)
)
=
Φ
f
(
Λ
)
Φ
τ
𝔵
in
,
in which ⊙ is the element-wise product, ƒ(Λ) is a general function based on the eigen-value matrix Λ, and Φ is the eigen-vector matrix.
7 . The method of claim 5 , wherein a convolution is defined as:
𝔵
o
u
t
=
∑
m
=
0
K
Φ
m
T
m
(
L
t
e
t
)
𝔵
i
n
where θ m are a set of learnable model parameters denoting the coefficients of the polynomials, and T m ∈Rn+n is the Chebyshev polynomial of order k.
8 . The method of claim 1 , wherein a lumped discrete LBO on Tis defined as:
Δ
f
(
v
i
,
)
=
1
d
i
∑
j
∈
N
(
i
)
k
i
,
j
(
f
(
v
i
)
-
f
(
v
j
)
)
where N(i) includes the adjacent vertices of vertex v i , and d i is the total tetrahedral volume of adjacent tetrahedral to vertex v i , and k i,j is the string constant.
9 . The method of claim 8 , wherein the tetrahedral mesh is decimated by an order of two using the function:
d
(
v
i
,
v
j
)
=
-
A
i
,
j
(
1
D
i
i
+
1
D
j
j
)
.
10 . A method, comprising:
computing a volumetric Laplace Beltrami operator (LBO) for a tetrahedral mesh, wherein computing the volumetric LBO is performed based on the following function:
Δ
f
(
v
i
,
)
=
1
d
i
∑
j
∈
N
(
i
)
k
i
,
j
(
f
(
v
i
)
-
f
(
v
j
)
)
where N(i) includes the adjacent vertices of vertex v i , and d i is total tetrahedral volume of adjacent tetrahedra to vertex v i , and k i,j is the string constant.
11 . The method of claim 10 , further comprising feeding the volumetric LBO into a neural network.
12 . The method of claim 11 , further comprising feeding a set of signals for each vertex of the neural network.
13 . The method of claim 12 , wherein the set of signals comprises at least one input signal and at least one output signal.
14 . The method of claim 13 , wherein the at least one input signal is defined as:
x in ∈R N .
15 . The method of claim 13 , wherein the at least one output signal is defined as:
x out ∈R M .
16 . A method, comprising:
down-sampling tetrahedral mesh on a neural network, wherein down-sampling is performed per the following function:
d
(
v
i
,
v
j
)
=
-
A
i
,
j
(
1
D
i
i
+
1
D
j
j
)
.
17 . The method of claim 16 , wherein the tetrahedral mesh is decimated by an order of two using the function:
d
(
v
i
,
v
j
)
=
-
A
i
,
j
(
1
D
i
i
+
1
D
j
j
)
.
18 . The method of claim 17 , wherein each layer of the neural network includes a down-sampling size of ¼.
19 . The method of claim 16 , further comprising feeding each layer of the neural network into a pre-computed volumetric LBO.
20 . The method of claim 19 , further comprising computing the volumetric LBO based on the following function:
Δ
f
(
v
i
,
)
=
1
d
i
∑
j
∈
N
(
i
)
k
i
,
j
(
f
(
v
i
)
-
f
(
v
j
)
)
where N(i) includes the adjacent vertices of vertex v i , and d i is total tetrahedral volume of adjacent tetrahedra to vertex v i , and k i,j is the string constant.Join the waitlist — get patent alerts
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